Photonic band gap (PBG) structures are periodic dielectric structures that exhibit frequency regions in which electromagnetic waves cannot propagate. The idea for PBGs was first proposed by Eli Yablonovitch in 1987. The interest in PBGs arises from the fact that photon behavior in a dielectric structure is similar to the behavior of electrons in a semiconductor. The periodic arrangement of atoms in a semiconductor lattice opens up forbidden gaps in the energy band diagram for the electrons. Similarly in PBG structures, the periodic placement of dielectric "atoms" opens up forbidden gaps in the photon energy bands. This analogy can be easily seen in the Schrodinger equation (1) (see below) of a propagating electron wave in a potential, V(r) and equation (2), which is derived from the Maxwell's equations, for the electric field amplitude E(r) propagating a monochromatic electromagnetic wave of frequency .omega. in an inhomogeneous but nondispersive dielectric media as shown in the following equations: ##EQU1##
Here, m is the electron rest mass and .psi.(r) is the scalar wave function in equation (1). In equation (2), the total dielectric is separated as EQU .epsilon.(r)=.epsilon..sub..omicron. +.epsilon..sub.fluc (r), (3)
where .epsilon..sub.o is the average dielectric constant value and .epsilon..sub.fluc (r) defines the spatially fluctuating part. The latter plays a role analogous to the V(r) in the Schrodinger equation, and the quantity .epsilon..sub.o .omega..sup.2 c.sup.2 is equivalent to the energy eigenvalue E of the Schrodinger equation.
The idea of PBGs has led to the proposal of many novel applications at optical wavelengths, such as thresholdless lasers, single-mode light-emitting-diodes and optical wave guides. In addition, PBGs are already being used in the millimeter and microwave regimes, where the applications include efficient reflectors, antennas, filters, sources and wave guides. They have also found possible applications as infrared filters. As a result, they have been extensively studied in the last few years.
The PBG structures behave as ideal reflectors in the band gap region. Depending on the directional periodicity of these dielectric structures, the band gap may exist in 1-D, 2-D or all the three directions.
One of the unique features of PBG structures is their scaleable characteristic from microwaves to optical frequency. This can be explained better by going back to Maxwell's equations. Equation (2) listed earlier is derived from Maxwell's equations which can be rewritten in a magnetic field vector form as: ##EQU2##
where H(r) is the magnetic field vector. A new dielectric constant is defined, EQU .epsilon.'(r)=.epsilon.(r/s)=.epsilon.(r') (5)
where s is some scalar parameter. Basically the dielectric has been compressed or expanded by this scalar value s. Now defining a new variable, r'=sr and .gradient.'='/s, equation (4) can be rewritten as: ##EQU3##
which can also be written as ##EQU4##
Here, .epsilon.(r'/s)=.epsilon.'(r') and this allows return to the master equation with mode profile H'(r')=H(r'/s) and frequency .omega.'=.omega./s. If the mode profile is studied after changing the length scale by a factor of s, the old mode and its frequency simply needs to be scaled by the same factor. The solution of a problem at one length scale determines the solutions at all other scales.
The modes of photonic crystals can be tested at microwave frequencies with bigger dimension and because of scalability of the structure it is ensured that the electromagnetic properties will not change at optical frequencies with submicron dimensions.
Now studying the effect of change in the dielectric configuration, suppose that a new system has a dielectric constant .epsilon.'(r)=.epsilon.(r)/s.sup.2. Therefore, ##EQU5##
The harmonic modes of the system are unchanged but all the frequencies have been scaled up by a factor of s. For example, if the dielectric constant is multiplied by a factor of 1/4, the mode patterns are unchanged but the frequencies are doubled. So, by changing the dielectric constant or changing the dimensions of the structure, the electromagnetic properties can be scaled anywhere from microwave to optical frequencies. Similar to the impurity doping in a semiconductor, localized electromagnetic modes can be created in the band gap region of PBG structures by introducing defects that disturb the periodicity of the structure. This can be achieved by adding extra material to the crystal, which acts like a donor atom of a semiconductor. The defect gives rise to donor modes which have their origin at the bottom of the conduction band. A defect can also be introduced by removing a part of the material, thus creating states similar to the semiconductor behavior with acceptor atoms. Experiments have shown that the acceptor modes, acting like cavities, are of greater importance with their highly localized and single-mode cavity characteristics. In photonic crystals with defects, the transmission spectrum is changed by the presence of a narrow transmission peak within the band gap. Defect peaks with quality factors in the range of 1000-2000 have been experimentally demonstrated.
Much of the PBG research effort up to this point has focused on the use of purely dielectric material to construct the PBG structure. Metallic photonic band gap (MPBG) structures have received relatively little attention due to perceived problems relating to lossiness in the metal components.
However, MPBG structures do have some distinct advantages over their all dielectric counterparts, and these advantages have garnered MPBGs more attention recently. MPBG structures offer the potential of lighter weight, reduced size and lower materials and fabrication costs when compared to all dielectric structures. The use of metal can also lead to fundamentally different PBG characteristics. For an interconnected mesh structure, the stopbands of the MPBG will extend from zero frequency up to some cut-off frequency, which is determined by the periodicity of the structure. Such behavior is in contrast to purely dielectric PBG structures, which typically have stop bands extending over relatively narrow ranges of frequencies. On the other hand, it has also been shown that MPBG structures consisting of isolated metal patches have a band-stop behavior very similar to the all dielectric photonic band gap structures.
As stated above, the idea of photonic band gaps was first proposed by Yablonovitch in 1987. The idea is analogous to the behavior of electrons in a crystal lattice. The electromagnetic waves propagating in a structure with a periodically modulated dielectric constant are organized in "photonic bands" which are separated by "gaps" where propagating states are forbidden.
Following the inception of this idea, various lattice geometries were studied to find a periodic structure that would exhibit a photonic band gap in all the directions. After several unsuccessful attempts in finding the right lattice geometry using "trial-and-error" techniques, Ho et al. at Iowa State University were first to predict the existence of a complete band gap in a periodic dielectric structure arranged in diamond lattice geometry. Diamond lattice structures were calculated to have large gaps for refractive index ratio between the two dielectrics as low as two.
With these findings, Yablonovitch et al. fabricated the first three-dimensional photonic band gap structure 10 as illustrated in FIG. 1. The structure was arranged in a periodic face-centered-cubic lattice, but with cylindrical air holes 12, giving it an over-all diamond lattice structure. The periodicity in the structure was achieved by drilling holes 120.degree. apart and at 35.degree. from the z-axis into a dielectric slab as shown by rods 14 in FIG. 1. This experimental structure 10 exhibited a full 3-D photonic band gap. It had a forbidden gap from 13 to 16 GHz with 10 dB attenuation per unit cell.
However, the structure 10 has proven difficult to fabricate at optical frequencies where feature sizes are less than one micron. Chemically assisted ion beam etching technology has been used to etch the holes. It has been found difficult to maintain the linearity and hole size as the etching depth increases. This adversely affects the periodicity of the structure and hence the photonic band gap. The midgap optical reflectivity is found to be very sensitive to structural errors in the photonic crystal.
After the initial verification of the existence of photonic band gap, there was an increased effort to find new structures that could be more easily fabricated. Another structure that exhibited a complete band gap was suggested by Ho et al. and fabricated by Ozbay et al. from Iowa State University. This new "layer-by-layer" structure 16 illustrated in FIG. 2 was fabricated by stacking layers of equally spaced round or rectangular rods 18. The first layer-by-layer structure 16 was fabricated using alumina rods with dielectric constant .epsilon.=9.6 glued together to form a face-centered-tetrahedral symmetry. This structure 16 was found much easier to scale down in size. Standard silicon micro-machining techniques have been used to scale down the size of this layer-by-layer PBG structure 16 and make it operational up to 500 GHz at Iowa State University. Other laser prototyping methods have been used to fabricate this structure to allow operation to about 2 THz.
Fabrication of PBG structures has been mostly studied using frequency independent dielectric materials with positive dielectric constants where the possible problems related to absorption can be neglected. However, recent work on PBG structures using metals has shown good results for operating frequencies much lower than the plasma frequency of the metal. Although metals are quite lossy at optical frequencies, they act as nearly perfect reflectors at lower frequencies. The initial interest in metallic photonic band gap (MPBG) structures arose from the fact that they show higher attenuation with fewer layers as compared to their all dielectric counterparts. These structures are much more compact, lighter weight and can be fabricated at a reduced cost as compared to dielectric structures.
The first MPBG structure with a complete 3-D band gap was reported by Yablonovitch et al. in 1996. The MPBG structure 20 had a geometry resembling covalently bonded diamond as shown in FIG. 3A. Copper wire strips 22 shown in FIG. 3B snap together to make the diamond geometry. The transmission spectrum of this structure did show a forbidden gap similar to a purely dielectric structure centered about v.sub.o , corresponding to the lattice constant of the structure. But this forbidden gap does not extend in all the directions. In addition to this forbidden gap, this structure 20 shows a new 3-D band gap that extends from zero to a cutoff frequency of .about.(1/2)v.sub.o . This high pass spectrum of the structure can be observed in all the directions and is called the metallicity gap. The structure 20 shown in FIG. 3A has a high pass cutoff frequency of 6.5 GHz. The effect of introducing defects into the structure was also studied in detail. However, because of the complex geometry of the structure it has been difficult to scale down its size to allow operation at higher frequencies.
FIG. 4 illustrates another layer-by-layer structure 24 for metallic photonic crystal fabricated at Iowa State University by McCalmont et al. which has shown very good results. The structure 24 is built by stacking three layers of square metallic grids 26, 28, 30 aligned to each other and separated by a dielectric medium 32, 34 as shown in FIG. 4. The transmission characteristic of the structure 24 exhibits a metallicity gap with as few as three metal layers 26, 28, 30 in the stacking direction. The location of the band edge is a function of lattice constant, width of the metal wire grid, and the refractive index of the dielectric material. This structure 24 was fabricated on a duroid (.epsilon.r=1.5) printed circuit board 36 with copper laminations on both sides. One side of the copper cladding was patterned with the grid pattern 26 using standard photolithography techniques and the backside copper was completely removed. The structures were fabricated to operate in 75-110 GHz frequency range.
A change in periodicity of the center grid 28 creates a defect mode in the band gap region, effectively forming a bandpass characteristic. The size of the defect determined the location of the defect mode frequency in the band gap. The response of the structure is virtually unchanged over a wide range of incident angles, but it is not a fully three-dimensional structure. Both the transmission and reflection characteristic of the defect peak was measured. The reflected peak was much sharper than the transmission peak with a maximum measured quality factor of 461 in the reflection measurements.
McIntosh et al. studied a lattice geometry where metallic atoms were located on a three-dimensional oriented face-centered-cubic (FCC) lattice, and the sites were isolated from each other. Initially they studied the characteristics of this structure in the microwave region and then extended this concept to the infrared. The metallic FCC structure was embedded in a polymer and supported on a silicon substrate. This group fabricated two different sets of arrays.
The first set had its lattice sites occupied with square metal patches while the other set of arrays had circular patches. The lattice constants of different structures ranged from 2 to 4 .mu.m. These structures show band gap behavior that is very similar to the dielectric PBGs. The circular patch structures showed frequency gaps at 3650 cm.sup.-1 with a maximum attenuation of 15 dB and a gap to midgap ratio of 0.21. The circular patch had diameter, D 1.06 .mu.m and center-to-center spacing, S=1.7 .mu.m. The FCC structures with square patches at the lattice points had a maximum attenuation of 21 dB with center frequency of 1450 cm.sup.-1 and a gap-to-midgap ratio of 0.83. The metal patches were 2 .mu.m wide and the center-to-center spacing was 3.18 .mu.m.
These structures definitely have very high operating frequencies, but their attenuation is relatively low, and all of the measured values have been compensated by subtracting the silicon transmission spectrum. For this structure to be used as a filter, the silicon substrate would always be a problem. In addition, the band gaps in the samples of McIntosh et al. are also compensated by subtracting polyimide absorption. Separating the effects of the photonic band gap structure from the material absorption is problematic, and this overlap would preclude use of the structure from most applications.
The layer-by-layer MPBG structures described earlier are related to frequency selective surfaces (FSSs). Frequency selective surfaces are two-dimensional arrays of metallic aperture elements or patches that have frequency-filtering properties. They have been studied in great detail because of their application as filters, bandpass radomes, polarizers and mirrors in microwave and far-infrared region. Different aperture and patch element geometries (e.g. square patch, circular patch, cross dipole, Jerusalem cross, square loop, circular loop, square aperture etc.) have been studied.
Most of the experimental work reported on FSS has been for single layer structures, and no results have been reported for true three-dimensional structure. Single-layer free-standing copper grid structures 12 .mu.m thick have been reported with frequencies near the 2 THz, but the structure shows an attenuation of only 18 dB.
Metallic PBG structures on the other hand have much higher attenuation in the band gap region as compared to single layer FSSs. The metallic PBG is a three-dimensional structure which provides the unique advantage of creating a defect mode in the stop band region by disturbing the periodicity of the structure. Defect peak frequencies with very high quality factors have been reported. The layer-by-layer interconnected metallic structure discussed earlier show a quality factor, Q of 461. Quality factor is defined as the ratio of peak frequency to the 3 dB (half power) bandwidth of the peak. Also, the frequency of an MPBG defect peak is adjustable and is a function of the size of the defect introduced. Unfortunately, these devices have been difficult to construct to allow operation in the infrared region.